Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

TL;DR

This paper derives detailed asymptotic formulas for the Fredholm determinant of the sine kernel over unions of multiple intervals, extending Dyson's work to more complex geometries in the context of random matrix theory.

Contribution

It provides a rigorous proof of the second-order asymptotics for the sine kernel determinant over unions of intervals, generalizing previous single-interval results.

Findings

Asymptotics involve hyperelliptic integrals and Jacobi inversion problems.

Logarithmic derivative of the determinant has a linear and oscillatory component.

Asymptotics of the resolvent trace are also established.

Abstract

In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\sin(x-y)\over\pi(x-y)$ over this interval. A formal asymptotic expansion for the determinant as $s$ tends to infinity was obtained by Dyson. In this paper we replace a single interval of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect to $s$ of the determinant equals a constant (expressible in terms of hyperelliptic integrals) times $s$, plus a bounded oscillatory function of $s$ (zero of $m=1$, periodic if $m=2$, and in general expressible in terms of the solution of a Jacobi inversion problem), plus $o(1)$. Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.